Solid Geometry
Definition
Solid (or 3-dimensional) geometry refers to the study of objects like cubes, cones, cylinders, pyramids, and spheres, among others. Usually, we’re interested in calculating the surface area and/or volume of such an object.
Cubes
A cube is essentially a 3-dimensional square. Each die in a pair of dice is a cube, and so every cube has 6 sides, and each of its edges is the same length. The surface area of a square is simply 6 times the area of one of its sides:
\(\begin{align}A = 6 \times s^2\end{align}\)
The volume of a cube is even simpler, since it’s computed by multiplying length times width times height, but since each of these is the same with a cube, then the volume is given by this formula:
\(\begin{align}V = s^3\end{align}\)
Cones
We’re all familiar with ice cream cones, and a mathematical cone is simply an ice cream cone without any ice cream in it. In the figure, we show the cone upside down. As you can see, a cone’s base is simply a circle with radius equal to r. It then tapers to a point at the top, and the distance from the center of the circle to the apex of the cone is h.
The distance from the outside of the circle to the apex is s, and it can be calculated from r and h by using the Pythagorean theorem, since s is simple the hypotenuse of a right triangle whose legs have length r and h. That is,
\(\begin{align}s^2 = r^2 + h^2\end{align}\)
To calculate the cone’s volume, or in other words how much ice cream it would hold without overflowing, we use this formula:
\(\begin{align}V = \frac{1}{3} \pi r^2 h\end{align}\)
Cylinders
Cylinders are the essential elements in a car’s engine, and it’s easy to calculate both the surface area and the volume of a cylinder. As you can see in the figure, a cylinder is basically a can, with a circle at its base and at its top. The circle has radius r and the height of the cylinder is h. So the surface area can be easily found by taking the can apart and computing the area of its parts.
- There are two circles – one at the top and one at the base, and the area of each of these is \(\begin{align}\pi r^2\end{align}\). Therefore, the first part of the area formula is \(\begin{align}2 \times \pi r^2\end{align}\).
- To get the area of the vertical outside of the cylinder, we open up the can and take the circumference of the circle, \(\begin{align}2 \pi r\end{align}\) times the height of the can. This is \(\begin{align}2 \pi r h\end{align}\).
Adding these pieces together, we get:
\(\begin{align}A = 2 \times \pi r^2 + 2 \pi r h\end{align}\)
The volume of a cylinder is even simpler. To get it we simply take the area of one of the circles, and multiply it by the height:
\(\begin{align}V = \pi r^2 h\end{align}\)
Pyramids
A pyramid is similar to a cone, except its base is a polygon, like a square or triangle. We show two pyramids, one with a square base, and one with an equilateral triangular base. The volume of each is calculated the same way. It’s simply 1/3 the area of the base times the height. Or in other words, for the pyramid with a square base:
\(\begin{align}V = \frac{1}{3}s^2 h\end{align}\)
For the other pyramid it’s a little more complex, because the area of an equilateral triangle is not as simple as the area of a square. For any triangle, the area is 1/2 the base times the height of the triangle. If the side of an equilateral triangle is s, then its base is equal to s, and its height is equal to \(\begin{align}\frac{\sqrt{3}}{2}s\end{align}\). So its area is
\(\begin{align}\frac{1}{4} \sqrt{3}s^2\end{align}\)
And so the volume of the pyramid with the equilateral triangular base is:
\(\begin{align}V = \frac{1}{12} \sqrt{3}s^2 h\end{align}\)
Spheres
A sphere is a ball or globe or “3-dimensional circle.” Its radius r is the only measurement we need in order to calculate its surface area and its volume.
\(\begin{align}A = 4 \pi r^2\end{align}\)
\(\begin{align}V = \frac{4}{3}\pi r^3\end{align}\)
